The immersed interface method for axis-symmetric problems and application to the Hele–Shaw flow

Many physical application problems are axis-symmetric. Using axis-symmetric properties, many three dimensional problems can be solved efficiently using two dimensional axis-symmetric coordinates. In this paper, the immersed interface method (IIM) in axis-symmetric coordinates is developed for elliptic interface problems that have a discontinuous coefficient, solution or flux. A staggered grid is used to overcome the pole singularity. Other challenges include deriving the jump relations in axis-symmetric coordinates, designing the numerical algorithm when the interface is close to the pole (r = 0); computing interface quantities such as the normal and tangential directions, surface derivatives, curvature information, etc. The numerical algorithm is based on a finite difference discretization and uniform grid in the axis-symmetric coordinates. The finite difference scheme is the standard one away from the interface but is modified at grid points near and on the interface. The method is shown to be second order accurate in the infinity norm. The developed new IIM is applied to the Hele–Shaw flow and compared with the results from the literature.